Abstract
Exponentiated power Lindley distribution is proposed as a generalization of some widely well-known distributions such as Lindley, power Lindley, and generalized Lindley distributions. In this paper, the exact explicit expressions for moments of order statistics from the exponentiated power Lindley distribution are derived. By using these relations, the best linear unbiased estimates of the location and scale parameters, based on type-II right-censored sample, are obtained. Next, the mean, variance, and coefficients of skewness and kurtosis of some certain linear functions of order statistics are calculated and then used to derive the approximate confidence interval for the location and scale parameters using the Edgeworth approximation. Finally, some numerical illustrations and two real data applications are presented.
1. Introduction
Ashour and Eltehiwy [1] introduced the exponentiated power Lindley (EPL) distribution as a generalization of two-parameter power Lindley distribution and they studied some mathematical properties of the EPL distribution. They also showed that this distribution provides more flexibility to analyze a complex real data set. Due to its practicality, the EPL distribution can be used for many applications, including accelerated life testing, survival analysis, reliability, biology, and others. The EPL distribution has attractive feature that its hazard rate function could be decreasing, increasing, and decreasing-increasing-decreasing but not constant, depending on the shape parameter. The EPL distribution has a unimodal and right skewed probability density function (PDF) for . Many authors have developed generalization of Lindley distribution. Prominent among these generalizations which we are aware of are generalized Lindley distribution by Nadarajah et al. [2]; power Lindley distribution by Ghitany et al. [3]; generalized inverse Lindley distribution by Sharma et al. [4]; pseudo-Lindley distribution by Nedjar and Zeghdoudi [5]; size-biased gamma Lindley distribution by Beghriche and Zeghdoud [6]. Ashour and Eltehiwy [1] observed that in numerous situations, the EPL distribution provided a better fit than Lindley, power Lindley, generalized Lindley, exponentiated exponential, modified Weibull, and Weibull distributions.
Over the past six decades or so, several authors have shown keen interest in order statistics. These order statistics and their moments are of great significance in many real life applications involving data relating to flood, drought, reliability, engineering, and life testing situation. In recent times, several papers and books have been published on order statistics and their distributional properties. Among them are David and Nagaraja [7]; Arnold et al. [8]; Balakrishnan and Cohen [9]; Balakrishnan and Ahsanullah [10]; Balakrishnan and Sultan [11]; Malik et al. [12]; Mahmoud et al. [13]; Genc [14]; MirMostafaee [15]; Balakrishnan et al. [16]; Kumar and Dey [17]; Sultan et al. [18]; Sultan and AL-Thubyani [19]; Kumar et al. [20]; and so on. Due to the wide applications of order statistics, several authors have carried out extensive studies on these kinds of ordered data. Numerous papers dealing with moments and estimation of parameters of different lifetime distributions based on order statistics can be found in literature. For example, Balakrishnan and Cohen [9] discussed the moments and estimation of parameters. Sultan et al. [18] obtained the moments of power function distribution. Sultan and AL-Thubyani [19] developed the higher-order moments and inferential procedure for estimating parameters of Lindley distribution. Ahsanullah and Alzaatreh [21] studied the moments and inferential procedure for log-logistic distribution. Recently, Kumar and Goyal [22, 23] and Kumar et al. [24, 25] developed inferential procedure for power Lindley and generalized Lindley, modified power function, and generalized inverse Lindley distribution.
Let be an independent random variable from the EPL distribution with cumulative distribution function (CDF) and PDF is given, respectively, by
The hazard function of the EPL distribution is given bywhere
Some widely well-known distributions can be obtained as special cases from the EPL distribution. By setting and in both (1) and (2), the CDF and PDF of the generalized Lindley distribution and power Lindley distribution are obtained, respectively, while setting gives the CDF and PDF of Lindley distributions.
To the best of our knowledge, there are no reports on moments and estimation for the EPL distribution using order statistics. In this paper, first is to obtain the exact explicit expressions for moments of order statistics from the EPL distribution, and second is to derive the best linear unbiased estimates (BLUEs) of the location and scale parameters of the EPL distribution based on order statistics.
The rest of this paper is organized as follows: in Section 2, the moments of EPL distribution are derived based on order statistics. The BLUEs of the location and scale parameters are provided in Section 3. Next, these BLUEs are used in Section 4 to obtain the coefficient of skewness and kurtosis of some pivotal quantities. Also, we obtain Edgeworth approximations for the distributions of these pivotal quantities. Simulation study and real data analysis are presented in Sections 5 and 6. Finally, the paper ends with a conclusion in Section 7.
2. Moments of Order Statistics
In this section, the moments of order statistics are established for a given random sample from the EPL distribution.
2.1. Single Moments
In this subsection, the explicit expressions for single moments of order statistics are derived. Let be the order statistics for a given random sample from the EPL distribution, and , , denotes the th order statistics with PDF which is given by (see [7])where and are given by (1) and (2), respectively, and
Theorem 1. For the EPL distribution given in (1), the th moment of the th order statistics , and , is as follows:
Proof. From (5), we haveBy using (1) and (2), one can writehence the result.
Some special cases from equation (7):(1)If , we get(2)If , we getas obtained by Kumar and Goyal [23].(3)If and , we get(4)If and , we get(5)If and , we get(6)If and , we getas obtained by Ashour and Eltehiwy [1].
2.2. Double Moments
Let be the order statistics from the EPL distribution. Then, the joint PDF of the th and th order statistics is given aswhere and .
Theorem 2. For the EPL distribution given in (1), the double moment of th and th order statistics, , , is as follows:
Proof. Using (16), we haveBy using the same argument as in Theorem 1, we get the result given in (17).
2.3. Triple Moments
Let denote the order statistics drawn from the EPL distribution. The joint PDF of the th, th, and th order statistics is given bywhere and
Theorem 3. For the EPL distribution given in (1), The triple moment of th, th, and th order statistics , , is as follows:
Proof. From equation (19), we haveBy using the same argument as in Theorems 1 and 2, we obtain the result given in (21).
2.4. Quadruple Moments
Let denote the order statistics drawn from the EPL distribution. The joint PDF of the th, th, th, and th order statistics is given aswhere and
Theorem 4. For the EPL distribution given in (1), the quadruple moment of th, th, th, and th order statistics is as follows:
Proof. From equation (23), we haveBy using the same argument as in Theorem 3, we get the result given in (25).
3. Estimation of Parameters
Here, we study parameter estimation for the EPL distribution based on order statistics.
3.1. BLUEs of Parameters
The PDF of the scale-parameter EPL distribution isand the PDF of the location-scale parameter EPL distribution is
Let , denote type-II right-censored sample from the location-scale parameter EPL distribution in equation (1). Let us denote , , , and , . Therefore,where , and , and . Then, the BLUEs of and can be computed as follows (see Arnold et al. [8]):where
The variances and covariance of these BLUEs can be computed as follows (see Arnold et al. [8]):
The values of and are displayed in Tables 1 and 2 for different values of sample sizes and different censoring cases and for some selected values for . The coefficient of the BLUEs and are given by (17) and (18), respectively, with conditionswhich are used to check the computations accuracy.
4. Approximate Inference
Here, we derive the confidence intervals for the parameters and based on the following pivotal quantities:where and are the BLUEs of and with variances and , respectively. is used to draw inference for when is known, while can be used to draw inference for when is unknown. Similarly, can be used to draw inference for when is unknown.
The moments presented in Section 2 are used to derive the confidence intervals of the location and scale parameters based on the pivotal quantities in equation (35).
Hence, and can be rewritten aswhere , , is the standardized form of the available type-II right-censored sample , .
we consider finding the approximate distribution by using Edgeworth approximation for a statistic S (with mean 0 and variance 1) aswhere and are the coefficients of skewness and kurtosis of , respectively, and and are the CDF and PDF of the standard normal distribution, respectively.
5. Numerical Aspects
The relations obtained in the preceding sections allow us to evaluate coefficients of the BLUEs. The coefficients of the BLUEs are presented in Tables 1 and 2 while the variances and covariances of the BLUEs are presented in Table 3. From the results presented in Table 3, we observe that the variance of the BLUEs increases as the censoring level increases while the variance of the BLUEs decreases when the sample size increases and increases as increases. In addition, we observe that the covariances of the BLUEs decrease as the censoring level increases while the covariances of the BLUEs increase when the sample size increases and decrease as increases.
Tables 4–6 display the simulated percentage points at and sample size . Table 7 presents the values of the mean, variance, and coefficient of skewness and kurtosis and . The simulated average widths of confidence intervals are presented in Table 8. We observe that the Edgeworth approximations of the distributions of and both work quite satisfactorily; this is also clear from the average width of the confidence intervals based on and which are presented in Table 8.
6. Real Data Analysis
In this section, we analyze two real data sets to show the importance of the proposed estimators. One data set from environmental monitoring and another data is from maximum flood level for the Susquehanna River at Harrisburg.
Example 1. Analysis of clean upgradient ground-water monitoring wells subjected to mg/L.In the first data set, we consider vinyl chloride data obtained from clean upgradient monitoring wells which was studied by many authors such as Bhaumik and Gibbons [26]; Krishnamoorthy et al. [27]; Bhaumik et al. [28]; and Kumar and Goyal [22, 23]. The data are as follows:Now, a random sample of size 10 is selected from the given data set 1.0, 1.2, 3.2, 2.4, 0.8, 2.0, 0.4, 0.2, 2.9, 1.2. By the EPL distribution for the given sample, we have obtained the maximum likelihood estimate of , , and . By computation, the Kolmogorov–Smirnov (K–S) distance and the corresponding values are 0.163772 and 0.9513, respectively, which implies that the EPL distribution provides a reasonable model for this data. Moveover, the empirical cumulative distribution function (ECDF) plot and the Quantile-Quantile (Q-Q) plots are also presented under the vinyl chloride data as shown in Figure 1, which also imply that the EPL distribution can be used as a proper model to fit this data.
Then, by using the BLUE coefficients in Tables 1 and 2, we have
(a)
(b)
Example 2. Analysis of the maximum flood level for the Susquehanna River at Harrisburg, Pennsylvania.The second data set presents the maximum flood level for the Susquehanna River at Harrisburg, Pennsylvania, and it was studied by Dumonceaux and Antle [29]. The data areNow, a random sample of size 10 is selected from the given data set 0.418, 0.265, 0.613, 0.269, 0.654, 0.338, 0.315, 0.379, 0.297, 0.412. By the EPL distribution for the given sample, we have obtained the maximum likelihood estimate of , , and . By computation, the Kolmogorov–Smirnov (K–S) distance and the corresponding values are 0.14282 and 0.9694, respectively, which implies that the EPL distribution provides a reasonable model for this data. Moreover, as a further illustration, the empirical cumulative distributions plot and overlay the theoretical EPL distribution, the quantile-quantile (Q-Q) plots are also presented under the maximum flood level data, as shown in Figure 2, which also imply that the EPL distribution can be used as a proper model to fit this data.
Then, by using the BLUE coefficients in Tables 1 and 2, we have
(a)
(b)
7. Conclusion
In this paper, we have considered the EPL distribution when data are available in the form of order statistics. We first presented expressions for the single, double, triple, and quadruple moments. By using these moments, we have calculated the BLUEs of the location and scale parameters and the coefficient of skewness and kurtosis for some linear pivotal quantities. In the simulation study, we observed that the variance of the BLUEs increases when a high censoring level is taken into account; however, it decreases when a large value of parameter is considered corresponding to a fixed value of parameters and . Also, the covariance of the BLUEs decreases as the censoring level increases while it increases when the sample size increases and it decreases as increases. We next considered the distributions of the pivotal quantities in terms of Edgeworth approximation. These pivotal quantities are used to construct the interval estimation for the location and scale parameters. From our finding, we see that the moments of order statistics of the distribution are well behaved. This will encourage the study of the other properties of order statistics for a future research.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group no. RG-1438-086.